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Aspects of discontinuous Galerkin methods for hyperbolic conservation laws. (English) Zbl 0996.65106
Summary: We review several properties of the discontinuous Galerkin method for solving hyperbolic systems of conservation laws including basis construction, flux evaluation, solution limiting, adaptivity, and a posteriori error estimation. Regarding error estimation, we show that the leading term of the spatial discretization error using the discontinuous Galerkin method with degree $$p$$ piecewise polynomials is proportional to a linear combination of orthogonal polynomials on each element of degrees $$p$$ and $$p+1.$$ These are Radau polynomials in one dimension. The discretization errors have a stronger superconvergence of order $$O(h^{2p+1}),$$ where $$h$$ is a mesh-spacing parameter, at the outflow boundary of each element.
These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors in regions where solutions are smooth. We present the results of applying the discontinuous Galerkin method to unsteady, two-dimensional, compressible, inviscid flow problems. These include adaptive computations of Mach reflection and mixing-instability problems.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 76M10 Finite element methods applied to problems in fluid mechanics 76N15 Gas dynamics (general theory)
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